UC-NRLF 


PROPORTIONS 


lifts. 


N  BRIDGES. 


r,v 


CHAELE8    1VWNDER,    0.  E., 

M13MRKII    Or    TiIK   AMRR.v  AX    .-'iJiKTY    OF    <'iVir,   KNT(i(.VRKKS,    AND 
OF    TIII5    (JKMMAN    SOCIKTV    OF    K\(HM  RKS    IN    15KKUN. 


W    YOBK: 

CRAND,   PUBLISHER, 


r  AND  27 


1873 


N  STRRRT. 


••8a 


>H 

CZ 

< 

K 

m 


;  O 


V? 


OF 


PINS  Usm  IK  BRIDGES. 


CHA 


C.E., 


MEMBER    OF    THE   AMERICAN    SOCIETY  OF    CIVIL  ENGINEERS,   AND 


OF  THE  GERMAN  SOCIETY  OF  ENGINEERS  IN  BERLIN. 


NEW    YOKK: 
D.  VAN  NOSTRAND,  PUBLISHER, 

23  MURRAY  AND  27  WARREN  STREET. 

1873. 


Entered  according  to  Act  of£Congress,[in  the  year  1873,  by 

D.  VAN:NOSTKAND, 

in  the  Office  of  the  Librarian  of  Congress,  at  "Washington. 


PROPORTIONS 


Pins  Used 


A  portion  of  this  work  without  the 
mathematical  deductions  was  presented  to 
the  American  Society  of  Civil  Engineers, 
February  19th,  1873. 

It  is  not  the  least  of  the  merits  of  skele 
ton  structures,  such  as  are  built  by  the  best 
American  constructors,  that  any  of  the  parts 
can  be  exactly  calculated.  The  determina 
tion  of  strains  acting  in  pins  is  not  excepted 
from  this  statement,  though  it  is  connected 
with  the  application  of  some  of  the  finer 
laws  of  the  theory  of  elasticity. 

All  rules  referring  to  the  size  of  pins  can 


4 


be  immediately  deduced  from  the  few 
physical  principles  on  which  the  theory  of 
elasticity  is  based,  and  without  new  experi 
ments.  On  the  contrary,  sound  practice 
would  have  further  advanced  and  mistakes 
been  avoided,  if,  at  least,  a  scientific  exami 
nation  of  the  subject  had  been  undertaken 
previous  to  experiment,  in  order  to  arrive 
at  a  distinct  opinion  about  what  confirma 
tion  by  test  was  required  of  the  theory  of 
elasticity.  There  are  three  questions  to  be 
examined  in  the  present  paper,  viz.  : 

1.  What  is  the  law  of  distribution  of  the 
pressure  caused  by  the  tie  bar  on  the  bear 
ing  surface  of  the  pin. 

2.  What  is  the  law  of  distribution  of  the 
shearing  strain  over  the  cross-section  of  the 
pin,  and 

3.  What  is  the  value  of  its  bending  mo 
ment  in  any  of  its  sections. 

The  solution  of  the  first  actually  includes 
also  that  of  the  third,  but  it  is  more  con 
venient  to  treat  the  two  separately. 

For  this  purpose,  consider  a  top  chord  pin 
of  a  truss  bridge,  represented  by  Fig.  1. 


The  originally  straight  pin  M  M  is  rest 
ing  on  bored  bearing  surfaces  N  N,  of  a  top 
chord  casting,  and  carries  two  flat  eye  bars 
of  the  width  "  b  "  and  the  thickness  "  t  " 
placed  outside  of  the  casting.  This  is  the 
arrangement  of  any  pinned  bottom  chord, 
or  the  chains  of  a  suspension  bridge,  where 
the  outermost  bars  produce  strains  analogous 
to  those  borne  by  the  parts  here  shown. 

Under  strain,  the  pin  M  M  will  be  press 
ed  in  to  the  bearings  N  N,  which  are  sup 
posed  to  be  of  a  length  B  C=l,  and  of  the 
depth  d.  Both  bearing  surfaces  E  E,  be 
tween  pin  and  bars,  will  be  compressed, 
and  a  greater  share  of  the  total  pressure 
absorbed  by  the  parts  of  the  pin  closest  to 
the  face  of  the  casting.  The  pressure  per 
square  unit  at  E  will  be  greater  than  at  E, 
and  likewise  the  pressure  at  B  than  at  C. 
The  pin  is  supposed  to  accurately  fit  into 
its  bearings,  and  (for  the  present)  friction 
between  the  cylindrical  bearing  surfaces  is 
disregarded. 

So  far  as  the  half  cylindrical  bearing  sur 
face  or  half  circle  of  any  cross  section  of  the 
pin  is  concerned,  the  pressure  will  not  be 


6 


any  more  uniformly  distributed  than  is  the 
case  for  the  different  points  of  lines  E  F 
and  B  C.  The  pressure  of  a  pin  with  play 
first  will  be  concentrated  at  a  point,  but 
under  the  load  a  set  will  take  place,  and 
the  pressure  be  more  uniformly  distributed 
over  the  diameter  r  r.  The  law  of  this 
distribution  is  very  complex,  and  will  not 
be  followed  up  here.  It  is  sufficient  for  the 
present  to  consider  the  pressure  as  being 
uniformly  distributed  over  the  diameter  of 
the  pin,  which,however,is  somewhat  less  than 
is  actually  the  case.  The  diameters  of  pins 
found  under  this  supposition  will  be  a  little 
too  small;  reasons  will  be  given  for  as 
suming  this  basis  of  the  calculation. 

It  may  be  added  that  the  strain  per 
square  inch  will  not  only  decrease  in  a 
single  ratio  with  the  increase  of  the  pin, 
but  that  a  larger  pin  will  cause  a  more  uni 
form  distribution  of  the  pressure ;  the  im 
portance  of  the  least  practicable  play  in  the 
pin  hole  is  evident. 

The  pressure  per  square  inch  is  in  direct  pro 
portion  to  the  amount  of  depression  ;  conse 
quently  the  problem  is  the  same  as  to  find  the 


PH 


8 


curve  of  depression.  The  curve  itself  consists  of 
three  parts  ;  the  central  part  H  H  is  circular,  be 
cause  the  moment  of  flexure  of  H  and  H  is  of  a 
constant  value  ;  the  part  G  H,  which  differs  from 
H  H  as  well  as  from  M  G,  will  be  examined  first. 
Take  the  origin  of  the  co-ordinates  in  the  surface 
B  F,  in  the  original  centre  line  of  the  pin,  which 
at  G  has  received  a  depression  Y  o — and  for  any 
subsequent  abscisses,  has  a  deflection  y. 

The  deflections  y,  are  decreasing  towards  H,  and 
consequently  the  first  differential  will  have  a  nega 
tive  value. 

The  cross  section  (Fig.  2)  of  the  pin,  represents 
how  the  deflection  y  is  composed  of  two  parts : 
one  due  to  the  compression  of  the  pin  itself,  the 
other  to  the  compression  of  the  bearing,  and  fit 
ting  to  each  other.  These  partial  depressions  are 

"7T-  .  -~-  and  d  .  -=-,  where  Ax  is  the  pressure  per 

«a  jjj  Ki 

square  inch,  and  E  and  E  are  the  moduli  of  ten 
sile  elasticity  of  the  material  of  which  respectively 
the  pin  and  bearing  are  composed. 
The  total  depression  consequently  will  be  : 

M  Mi  =  O  p+p  X  =  O  X  =  (^~  +  -f-)±  x 

E 


9 


E 

When  th«  bearing  is  of  cast-iron  -^-  can  b«  as- 


sumed  to  be  about  =  ~,  so  that 

A 


d 


and  if 


is  represented  by  L 
There  will  be  the  equation 

A  x 
y  =  -rr.L    . 

£j 


E 


.     Equation  I , 


10 


This  equation  also  holds  good  for  the  pin-end 
M  G,  where  L  has  only  to  be  replaced  by  a  proper 
part  of  the  length  of  the  bar. 

The  pressure  Ax  multiplied  by  the  width  of  the 
bearing,  represents  the  increment  of  the  shearing 
force,  taken  per  unit  of  the  abscissa  ;  it  therefore 
depends  simply  on  the  value  y,  and  varies  with 
the  abscissa  ;  the  law  of  the  curve  will  be  repre 
sented  by  an  unknown  function  of  y  and  X. 

The  fundamental  law  of  any  flexure  of  a  beam  is 
expressed  by  the  equation 


which  expressed  in  words  is  —  the  product  of  the 
modulus  of  elasticity  E,  the  moment  of  inertia  I, 
of  the  section  of  the  beam  (here  a  pin),  and  the 
second  differential  coefficient  of  y  is  equal  to  the 
moment  of  exterior  forces  Mx  taken  at  the  point  of 
abscisses  considered. 

.  It  is  known  from  the  elementary  theory  of  elas 
ticity  that  the  first  differential  coefficient  of  Mx  is 
the  shearing  force  for  the  point  whose  abscissa  is 
x,  and  that  its  differential  coefficient—  the  second 
differential  coefficient  of  Mz  is  the  increment  of 
the  shearing  strain. 

This  increment  can  be  derived  from  equation  L, 

•pi  j 
which  gives  A*  .  d  =  y  .  —  —  -,   whence  by  taking 

M-* 

the  second  differential  of  equation  II.,  there  is 

found 

d^/       d*M,  Ed 

d*  4  ~       d*  »  "  y  '   L 


11 

or 

d4  y" 


IIL 


which  integrated  will  give  the  equation  of  the  curve. 
This  is  a  lineal  differential  equation  of  the  fourth 
order,  and  when  _ 

d  •  11  *    v      4  n*~ 

T  —  is  called  p4,  where  p  =     /  =-j- 

it  will  lead  to  the  general  formula  :  there  being 
e  =  2.7182818. 

y  =  Ax  eps+An  e-^+Bj  cosp  x-\-'Bll  sinpaj.  IV. 
for  which  the  4  constants  must  be  determined  to 
suit  the  conditions  of  the  piece  G  H  of  the  pin. 
These  are  : 

(a.)  For  x  =  0  or  at  point  a  the  moment  of  flex 
ure  is  a  known  quantity  =  P  z  so  that 

d2  11 
EI-j-f  =p2E  I  [A!  eps+A-n  e-^-B,  cospcc 


or  by  inserting  cc=  0  in  the  formula 


(6.)  For  x  =  0  the  shearing  force  is  equal  to  P, 
so  that 


andA1-A11-Bll  = 

(c.)  At  the  end  of  the  bearing  the  whole  pres 
sure  P  has  been  absorbed  by  the  bearing  casting, 
so  that 


12 

for  x  =  I  = 


d  *C 

B,  sinpZ-Bj!  cospM  =  0,  and 

A,**1  —  Alle-Pl  +  BA  sin  pZ-Bn  cospZ  =  0. 

(d  )  The  angle  under  which  the  tangent  of  the 
curve  at  H  will  cut  the  line  of  abscisses  can  be 
calculated  thus  : 

The  piece  H  H  is  part  of  a  circle,  whose  highest 
point  is  midway  between  H  and  H1}  and  if  B  rep 
resents  the  radius,  then  will 

EI^-==p*  E!(A!  e*>l  +  AJt  e~Pl  -Bt  cospZ- 
xt 

B  1  1  sin  p  I) 
and  the  tangent  of  the  angle  in  question,  since  B 

TT    TT 

is  many  times  greater  than  —  —  =  a,  will  be  found 

SB 

==-—-,  this  being  the  absolute  value  of  the  tan- 

A 

gent,  whilst  the  first  differential  coefficient  —  -•   is 

a  x 

negative  (as  has  been  already  stated)  ;  then  will 
apz  (AL  e^l-j-Ai:L  e-?1—  Bx  cos  p  I—  B^  sinpZ) 

==p  (Aj  e?1  -An  e~-Pz  —  B!  sinp  Z  +  Bll  cos  p  0 
and 


—  B,  t  (ap  sinpZ— 
The  four  equations  a,  6,  c,  and  d  are  sufficient  to 
calculate  Alf  A^  Bt  and  Bllf  so  that  equation 
IV.  is  fully  developed,  and,  according  to  equation 


13 


I.  the  pressure  in  any  point  of  the  bearing  B  0,  can 
be  found. 

This  somewhat  intricate  investigation  was  neces 
sary  to  get  an  idea  how  the  pressure  P  will  be  dis 
tributed.  The  subsequent  example  will  illustrate 
the  theory  of  this  — 

Let  P  =3"  X  1"  X  10000  Ibs.  ==  30000  Ibs. 

The  lever  z  is  very  nearly  ^  of  tht  thickness  of 
the  bar  =  ^".  The  diameter  of  pin  =  3",  so  that 
1=4.  L  can  be  assumed  to  be  =  4",  I  =1",  E  = 
30000000  Ibs.  and  a  =5". 

The  value  p  =*  y'  J^-  =  0.  658  and  p  I  =  0.  658. 


r  -  -  z  '  p  300QO 

~£TE1  ~  2X0.658*X3UOOOOUOX4~ 

p  30000 

~ 


0.65S*X3000UOOOX4 

1.931. 
8  =  0.5179. 
sin  0.658  =  0.6115. 
cos  0.658  =  0.7912. 

And  these  values  put  in  the  equations  a,  5,  c,  d, 
lead  to 

Bt  =-0.000902 
B1X  =-0.000165 

A!  =-0.0000493 
AM  =-0.0006627. 

For  x  =  0  there  are 

2/o=  A!+  A,  1+B1=C+.2B1=-0.  001615"  and 


14 


For  x  =  I  —1"  there  are 

y  Z=- 0.001 252"  and  A  z=1,,=  9391  Ibs, 

If  the  pressure  in  the  pin  hole  were  all 
the  constructor  has  to  provide  for,  the  di 
mensions  of  the  pin  and  eye-bar  might  be 
determined  in  several  ways.  In  reality, 
these  dimensions  are  almost  fixed,  for,  as 
will  be  shown,  to  reduce  the  flexure  of  the 
pin,  the  bearing  surface  must  be  short. 
The  rule  may  be  adopted  to  make  the 
bearing  surface  B  0  as  long  as  the  eye-bar 
is  thick.  In  this  case,  the  pressure  at  B 
will  be  12,115  Ibs.  whilst  at  0  it  is  only 
9,391  Ibs.  per  square  inch  for  a  3"  pin 
acted  upon  by  a  3"  X  1"  eye-bar  strained 
to  10,000  Ibs.  per  square  inch. 

The  maximum  pressure  is  21  per  cent, 
greater  than  it  would  have  been  if  uni 
formly  distributed.  The  depression  at  B  is 
0.0016"  and  at  0  0.0012",  showing  the  in 
fluence  of  a  curvature  which  hardly  can  be 
measured  even  by  very  fine  tools,  and  gene 
rally  would  escape  notice.  The  rise  in  the 
centre  of  the  pin  between  H  and  H  will  be 


15 


less  than  5  X  (0.0016—0.0012)  =  0.002" 
— hence  it  is  unnecessary  to  provide  an 
upper  bearing  in  the  centre  of  the  casting 
if  the  pin  is  nearly  of  the  proper  propor 
tion  ;  the  play  in  the  pin  hole  usually  ex 
ceeds  •£%  of  an  inch,  which  is  more  than  six 
times  greater  than  the  rise  of  pin  between 
H  and  H. 

The  maximum  pressure  for  the  standard 
bearing  length  equal  to  one  thickness  of  the 
eye-bar  was  by  the  foregoing  calculation 
12,115  Ibs. ;  for  a  badly  fitting  pin  it  would 
be  much  larger,  since  then  it  is  not  uni 
formly  distributed  over  the  diameter  of  the 
pin,  but  concentrated  at  one  point.  But 
for  a  well  fitting  pin  of  large  dameter  the 
pressure  of  12,000  Ibs.  per  square  inch  is 
not  too  large ;  and  for  simplicity,  it  is  well 
to  assume  that  this  pressure  is  uniformly 
distributed  over  the  diameter  of  the  pin, 
until  at  least  the  effect  of  "play  "in  the 
hole  has  been  directly  determined  by  a 
large  number  of  experiments  on  impact. 

The  later  experiments  prove  conclusively, 
that  wrought-iron  after  millions  of  impacts 
may  break  on  the  side  where  the  strain  is 


16 


tensile,  but  never  on  the  side  where  the 
strain  is  compressive.  Experiments  recent 
ly  made  in  this  country  as  to  the  crushing 
strength  of  wr ought-iron  support  this  obser 
vation,  the  ultimate  crushing  strength  hav 
ing  reached  60,000  Ibs.  per  square  inch. 
This  quite  disproves  conclusions  from  older 
experiments  carried  up  to  ultimate  strength, 
which  led  to  the  belief  that  iron  under 
compression  is  weaker  than  under  tension ; 
which  may  perhaps  be  true  for  very  soft 
metal.  But  such  iron  would  not  show  the 
same  behavior  when  used  in  a  bridge. 
A  properly  proportioned  bridge,  having  no 
section  strained  to  more  than  10,000  Ibs. 
per  square  inch,  will  never  break  from 
softness  of  metal  in  compression,  although 
it  may  after  the  passage  of  millions  of 
trains  by  the  ultimate  failure  or  wearing 
out  of  its  tension  members.  This  view  was 
always  held  by  the  best  engineers  on  the 
Continent  of  Europe,  and  General  Morin 
repeatedly  expressed  the  opinion  laid  down 
here. 

Fortunately  skeleton  bridges  such  as  are 
built  by  reliable  and  experienced  American 


^S/'JL  , 

engineers,  can  be  calculated ;  and  "all  good 
ones  are  calculated  so  that  no  detail  is 
strained  nearly  to  the  limit  of  durability. 

To  prove  how  specially  erroneous  are 
conclusions  derived  from  Hodgkinson's  ex 
periments  which  refer  to  alleged  differences 
in  the  compressive  strength  of  wrought  and 
cast  iron,  results  obtained  in  Prance,  and 
illustrated  by  General  Morin,  may  be  quo 
ted. 

Two  cast-iron  beams  were  made  of  the 
same  metal  and  with  the  same  height, 
length,  and  area.  One  of  these  beams  was 
constructed  to  suit  Hodgkinson' s  experi 
ments  with  a  heavy  tension  and  a  light 
compression  flange,  the  two  being  in  the 
ration  of  4-7  to  1 ;  the  other  had  two  equal 
flanges,  the  web  was  in  both  beams  of  the 
same  thickness.  The  Hodgkinson  beams 
deflected  2f  times  more  than  the  plainer 
one  ;  this  ratio  according  to  the  theory  be 
ing  exactly  as  that  of  the  moments  of  iner 
tia  of  the  cross  sections. 

The  Hodgkinson  beam  had  to  stand  pres 
sures  more  than  5  times  greater  than  those 
of  the  other  beam.  The  same  results  were  ar- 


18 


rived  at  by  testing  rolled  wrought-iron  beams 
with  unequal  flanges,  which,  when  reversed, 
gave  precisely  the  same  deflections.  Thus, 
these  two  experiments  proved  that  the  rules 
which  Mr.  Hodgkinson  drew  from  tests  up 
to  ultimate  strength  were  useless,  to  say  the 
least,  for  parts  strained  below  the  elastic 
limit  of  the  material.  Again,  to  show  how 
unreliable  are  rules  derived  from  experi 
ments  carried  up  to  ultimate  strength,  those 
of  Mr.  Fairbairn  on  a  riveted  girder  are  re 
ferred  to.  They  were  such  as  to  expose  the 
material  to  strains  nearly  in  the  same  man 
ner  as  for  a  railroad  bridge.  The  girder 
broke  under  a  strain  of  not  more  than 
18,347  Ibs.  per  sq.  in.  after  only  5,175  im 
pacts,  and,  of  course,  on  the  tensile  part. 

It  must  be  supposed  that  this  girder  was 
of  good  workmanship,  and  at  least  of  the 
average  quality  of  English  iron ;  and  we 
know  that  a  good  wrought-iron  bar  does 
not  break  under  strains  of  30,000  Ibs.  after 
130,000,000  of  impacts. 

The  girder  was  repaired ;  when  tested 
under  a  strain  of  13,000  Ibs.  per  sq.  in.,  it 
did  not  break  after  2,720,000  impacts.  In 


19 


accordance  with  new  experiments,  probably 
this  girder  would  have  broken  after  a  suffi 
cient  number  of  impacts,  still,  evidently 
from  the  first  experiments,  it  actually  had 
less  than  60  per  cent,  of  what  was  thought 
to  be  the  available  area. 

Since  in  such  girders  generally  about  20 
per  cent,  of  material  is  wasted  in  rivet  holes, 
it  may  be  said,  though  properly  designed 
and  constructed  according  to  the  rules  de 
rived  from  experiments  on  ultimate  strength, 
that  when  tested  in  the  same  manner  as  in 
practice,  they  have  lost  more  than  half  of 
the  value  of  their  metal.  The  failure  of 
the  Crumlin  Viaduct  superstructure  is  an 
other  and  more  direct  illustration  in  proof 
that  experiments  to  the  ultimate  strength 
cannot  be  relied  on  in  deducing  rules  for 
the  proportions  of  pins  in  bridges. 

This  failure  was  due  to  pins  proportioned 
by  a  rule  derived  from  experiments  on  the 
ultimate  shearing  strength  of  rivets.  Such 
rules,  applicable  to  boilers  or  ships  where 
the  ultimate  strength  must  be  taken  into 
account,  cannot  be  safely  used  to  determine 
the  proper  diameter  of  a  pin.  One  of  their 


20 


defects  is  that  no  allowance  is  made  for 
pressure  in  the  hole,  which  frequently  is  2  J 
times  the  strain  calculated  to  be  uniformly 
distributed  over  the  cross  section. 

Rivets,  on  account  of  friction  caused  by 
their  heads,  transfer  a  portion  of  this  pres 
sure  to  the  outer  surface  at  the  plates  they 
join  together,  and  therefore  do  not  give 
strikingly  bad  results  in  practice. 

The  same  rules  cannot  apply  to  pins 
w.here  no  such  pressure  on  the  surface  takes 
place. 

The  deduction  of  rules  for  pins  from  the 
condition  of  rivets  is  not  the  only  empiricism 
in  regard  to  the  former.  In  Germany,  on 
occasion  of  the  erection  of  suspension 
bridges,  Engineer  Malberg  made  trials  of 
links  which  gave  results  agreeing  with  those 
obtained  later  in  England.  Though  the 
German  proportions  were  published,  it 
seems  they  did  not  receive  attention  abroad, 
or  else  the  English  experiments  would 
probably  not  have  been  made.  As  far  as 
value  is  concerned,  neither  set  of  trials 
should  be  relied  on. 

Specifications   for   our    bridges    require 


21 


consideration  only  of  a  maximum  direct 
strain  under  the  heaviest  load  which  the 
structure  may  bear.  This  condition,  with 
reference  to  pins,  can  readily  be  fulfilled  by 
examining  analytically  the  nature  of  their 
strains.  A  pin  is  nothing  but  a  beam,  and 
since  a  great  variety  of  experiments,  made 
on  beams,  prove  that  within  the  limits  of 
elasticity  the  theory  adopted  is  not  less  cor 
rect  than  that  of  the  law  of  gravitation  to 
the  movements  of  the  planets,  what  re 
mains  to  be  done  is :  only  to  find  theoreti 
cally  the  maximum  strains  at  different 
points  of  the  pin.  Recourse,  however,  was 
had  to  empirical  researches,  and  tests  were 
made  which  could  not  show  the  nature  of 
the  strains  under  loads  such  as  occur  in 
practice.  Thus  the  first  experimental 
English  rule  made  no  allowance  for  pres 
sure  and  flexure,  but  referred  solely  for  the 
shearing  strain  which  was  supposed  to  be 
uniformly  distributed  over  the  cross  section 
of  the  pin.  It  is  plain  that  at  the  elastic 
limit  the  science  of  strains  ends,  since  this 
depends  -on  the  principle,  "  ut  tensio  sic 
vis." 


Nevertheless  this  is  still  frequently  over 
looked,  and  especially  so  in  reference  to  the 
shearing  strain  of  pins. 

The  rule,  moreover,  has  been  frequently 
misapplied  as  to  pins  of  suspension  bridge 
chains,  which  have  been  considered  as  sub 
jected  to  double  shearing,  although  the 
outer  bars  always  cause  but  single  shear 
ing,  and  also  not  unfrequently  to  bottom 
chord  pins,  where  likewise  the  outside  bars 
cause  but  single  shearing. 

This  rule  regarding  exclusively  the  shear 
ing  strength  of  a  pin  was  used  until  the 
failure  of  the  Crumlin  Viaduct,  and  ex 
perience  gained  with  suspension  bridges 
(such  as  built  at  Montrose  in  Scotland, 
where  the  pins  in  a  few  years  cut  their  way 
almost  through  the  eyes)  caused  engineers 
to  make  other  trials  referring  to  the  strength 
of  eyes  and  the  bearing  surface  of  pins. 
These  experiments  were  with  wide  and 
thin  bars,  as  used  in  suspension  bridges,  but 
not  in  truss  bridges,  of  good  design.  In 
this  case  the  eye  of  the  bar,  placed  between 
two  links  or  the  jaws  of  the  machine,  acts 
on  the  pin  by  double  shear ;  the  action  is 


23 


the  same  with  a  bar  as  wide  as  the  one 
tested  but  one  half  as  thick,  placed  outside 
of  the  bottom  chord  of  a  truss  bridge. 

The  new  rule  deduced,  fixing  the  diam 
eter  of  the  pin  from  f  to  f  of  the  width  of 
the  eye  bar,  contains  no  provision  for  the 
thickness  of  the  bar,  and  applies  to  the  case 
where  the  bar  is  20  times  as  wide  as  thick 
and  the  pin  is  subjected  to  single  shearing. 
Whether  this  rule  applies  to  a  square  bar 
is  more  than  doubtful  for  two  reasons.  First : 
Under  a  test  up  to  ultimate  strength  the  pin 
will  flatten,  the  bearing  surface  of  the  eye 
will  be  increased  and  receive  a  remarkable 
permanent  set,  and  the  now  tightly  fitting 
pin  will  exert  a  great  radial  pressure  on  the 
pin  hole,  which  causes  friction  that  may  be 
nearly  as  great  as  itself,  since  under  high 
pressure  friction  increases  greatly. 

A  pin  in  a  bridge  is  used  quite  differently  ; 
the  bearing  surface  is  much  less,  the  pin 
hole  will  flatten  but  little,  and  the  pin  would 
wear  out  the  hole  in  a  comparatively  short 
time  if  made  to  conform  solely  to  experi 
ments  on  the  ultimate  strength.  It  seems 
to  follow  from  such  experiments  that  the 


24 


pressure  on  the  bearing  surface  could  be 
considered  as  uniformly  distributed  over  the 
semicircular  surface  instead  of  through  the 
diameter  of  the  pin.  This  cannot  be  cor 
rect  for  a  pin  which  in  practice  has  a  play 
of  •ffV'h  to  ^jd  of  an  inch,  and  which  ought 
not  to  be  pressed  more  than  about  12,000 
Ibs.  per  sq.  in.  Second :  The  latest  Eng 
lish  rule  does  not  take  account  of  the  thick 
ness  of  the  bar,  and  of  the  moment  of  flex 
ure  to  which  the  pin  is  exposed.  This  has 
been  already  alluded  to,  and  is  really  the 
leading  point  in  determining  the  size  of  a 
pin ;  the  dimensions  which  satisfy  this  con 
dition  will  also  satisfy  the  two  others. 

Having  thus  explained  why  experimental 
reseaches  have  not  as  yet  established  practic 
able  rules  for  pins,  the  examination  of  shear 
ing  strains  with  reference  to  pins,  a  subject 
which  has  not  been  sufficiently  discussed  in 
many  text-books,  will  be  considered. 

SHEARING  STEAIN  IN  PINS. 

The  theory  of  flexure  teaches  that  the 
shearing  strain  is  not  uniformly  distributed 
over  the  cross  section;  consequently  the 


25 


maximum  shearing  strain  must  exceed  the 
strains  which  could  exist  if  uniformly  distrib 
uted.  To  show  the  amount  of  their 
difference :  Figure  3  represents  a  part  of  a 
bent  beam ;  C  E  and  D  Ex  must  be  two 
imaginary  sections  across  it,  and  V  V,  a  sur 
face  parallel  to  the  neutral  surface  G  B. 


The  question  is  what  forces  keep  the 
body  GDVYi  in  equilibrium.  The  mo 
ment  of  flexure  at  A  generally  differs  from 


26 


the  moment  at  B,  which,  in  this  examina 
tion,  is  supposed  to  be  the  greater.  The 
consequence  is  that  the  maximum  strain 
per  square  inch  at  D  is  greater  than  at  C. 
These  strains  per  square  inch  at  D  and  0 
may  be  represented  by  the  letters  Ax  and  A. 
The  strain  per  square  inch  in  any  point 

Y  is  AX  7r=r>  since  the  strains  decrease  in 

the  same  ratio  as  the  point  approaches  the 
neutral  line.  The  sum  of  all  the  strains  of 
the  surface  C  Y  is  partly  counteracted  by 
those  of  the  surface  D  Y,  the  last  named 
sum  being  the  greater.  It  consequently 
needs  a  shearing  force  S,  acting  in  the  di 
rection  of  A,  to  resist  the  forces  in  the  di 
rection  of  AX- 

Kepresent  by  V  any  distance  G  Y  and  by  u  the 
width  oo,  then  that  for  the  distance  V— VG  and  the 
total  shearing  strain  can  be  represented  oy 


If  the  sections  C  G  and  D  B  approach  each 
other  closely,  the  shearing  strain  must  be  con 
sidered  as  being  uniformly  distributed  over  the 
surface  W  and  the  strain  per  square  inch  will  be; 


27 


o  Ai      A.          1  / 

O  =  __  •  =  -r=  —  *  :  -  --  I  ( 
vv'  oo         vv       c  (x  .  u  «/ 


,     N 

vd  v) 


The  values  A  and  Al  can  be  derived  from  their 
respective  moments  M  and  Mx  so  that  when  I  rep 
resents  the  moment  of  inertia  of  the  cross-section 

of  the  beam  there  will  be  AX  =  -=-  .  Mx  and  A  = 

j-  .  Mj,  and  the  shearing  strain  per  square  inch 
will  be: 

I       M.-M      r 

a  — =F •  — == •  I  (u  v  d  u) 

I  .  u       vvi       J' 

v 

MT— M      dm     ^  ,  . 

and  — = — =-^ —  =  v  =  the  total  shearing  force 
vv1         d  % 

for  the  section  C  E  or  D  F,  so  that  the  shearing 
strain  per  square  unit  will  be 

a 


v      r 

-=—  .   I  (uvdu) 
lu     J 


For  v  =  0,  (j  will  be  a  maximum,  which  is  the 
case  for  the  neutral  line  itself,  whilst  the  integral 
decreases  to  nothing  at  the  point  D.  For  a  pin, 
the  section  is  a  circle,  and  u*-}-^  v*  =cZ8,  d  being 
the  diameter  of  the  pin,  and  the  maximum  shear 
ing  strain  will  be  found  : 

__4_      V 
=~3~(r27T) 

so  that  a  is  1 J  times  larger  than  if  the  total  shear- 


28 


ing  force  V  were  uniformly  distributed   over  the 
cross-section  r2  TT  of  the  pin. 

The  higher  and  more  accurate  examina 
tion  proves  that  the  shearing  strain  also 
is  not  uniformly  distributed  over  the  lines 
oo,  that  the  absolute  maximum  shearing 
strain  is  in  the  centre  of  the  pin,  and  is 
exactly  If  times  larger  than  it  would  be  if 
the  shearing  force  were  uniformly  dis 
tributed  over  the  whole  section. 

There  is  but  a  step  from  the  longitudi 
nal  to  the  vertical  shearing  strain.  That 
both  are  equal  in  every  point  of  the  beam, 
(on  the  pin)  can  readily  be  seen  by  con 
sidering  the  infinitely  small  hexahedron  of 
Figure  4. 

This  body  could  not  remain  in  equili 
brium  if  the  horizontal  shearing  force,  a, 
were  not  prevented  from  turning  the  figure 
Al3  CMD  around  point  B,  by  the  vertical 
shearing  force  a,  which  works  with  the 
lever  B  A,  to  turn  the  figure  A  B  C  D  the 
other  way.  A  B  being  ==  B  C  =  the 
units,  the  vertical  and  horizontal  shearing 
force  must  be  equal.  This  law  is  a  general 
one  suitable  for  any  body. 


29 


Now  it  has  been  proved  that  the  shearing 
maxima  strains  which  act  in  the  centre  of  a 
pin,  both  horizontally  and  vertically,  are 
If-  times  larger  than  if  it  were  possible  to 
distribute  the  total  shearing  force  V  uni 
formly  over  the  cross-section.  The  value 
V  reaches  its  maximum  just  at  the  facing 
of  the  casting,  and  is  equal  to  the  total 
tension  of  the  eye-bar. 


So 


FIG.  4. 

1    Ao              5 

)                                   € 
I 

1 

1 

j 

1 

/ 

I 

1 

1 

I 

1 

1 

I 

I 

i 
1 
i 
I 

1 
I 
1 

i 

1 

i 
i 

1 
/» 

How  large  can  the  shearing  strain  o  be, 
without  exceeding  the  ordinary  requirement 
for  iron  bridges,  that  no  tensile  strain  shall 
be  greater  than  10,000  Ibs.  per  square 
inch  ?  In  answering  this,  attention  is  called 
to  Figure  4. 

Shearing  strain  is  but  a  force  which 
tends  to  slide  an  infinitely  thin  slice  of  a 


30 


body  along  its  section,  as  for  instance,  the 
surface  A  B  parallel  to  C  D,  into  the  new 
position  A0  B0.  The  absolute  value  of  the 
emplacement  A  A0  depends  on  different 
causes : 

Firstly  on  the  shearing  strain  a  itself. 

Secondly  on  the  distance  A  D  between 
ATB  and  c  D,  for  a  surface  parallel  to  C  D, 
midway  between  A  and  D  would  only  slide 
half  as  far  as  A  B.  This  law  is  true,  since 
within  the  limits  of  elasticity  all  displace 
ments  increase  in  direct  and  single  ratio 
with  the  lengths. 

Thirdly  on  the  nature  of  the  material, 
so  that 

A  Ao=crXADX  (  -^-  )  where  I  -^-  J  is  a  co 
efficient  dependent  on  the  nature  of  the  material, 


or  it  is  ~-4^  =  -%-  tang .  angle  (ADA o). 

A.  JL)  \Jf 

Since  o  is  a  finite  value  like  G-,  the  angle 
A  D  A0  is  also  of  a  definite  value.  This 
angle  is  the  test  of  shearing  strain,  so  that 
wherever  an  angle  has  changed  from 
right,  a  pair  of  shearing  strains  must  have 
caused  it. 


31 


The  value  G  must  represent  a  weight  in 
order  to  make  --  an  abstract  fraction.  G 

(jT 

is  called  the  modulus  of  shearing  elasticity, 
and  represents  the  weight  which,  if  the 
limits  of  elasticity  would  reach  thus  far, 
were  sufficient  to  slide  a  surface  so  far  that 
the  angle  A  D  A0  would  become  45  deg. 

A  shearing  strain  can  always  be  resolved 
into  tensions  and  compressions  acting  in  all 
possible  directions  on  a  point  in  the  interior 
of  a  strained  body. 


.  5. 


In  Figure  5,  0  represents  a  point  of  a 
body,  A  AX  the  shearing  deplacement  of 
the  infinitely  near  point  A.  0  x  parallel 
A  A!  is  made  first  axis  of  the  system  of  co 
ordinates. 

The  original  line  0  A,  by  the  shearing 


32 


strain  has  been  changed  into  0  Ax,  which  is 
equal  to  OA  plus  AA0,  this  being  the  pro 
jection  of  A.A!  on  OA. 


The  shearing  strain  per  square  inch  being  <r, 
there  will  be 

A  &i=y  .  -77-  and  A  A  o  =  y.  -7^.  cos  a 

VJT  VJT 

and 

A  A  o        tension  y        a- 

"" 


or 


<  modulus  G 


E 

T  sin  oc  .  cos  or  .  -~-  a 


This  value  will  be  a  maximum  when  a 
=45  deg.,  so  that  Tension  maximum 

-i     E 

-*'~G~c'a 

For  a  perfectly  homogeneous  body  by 
experiment  and  calculation,  G  is  found  to 
=|  E,  so  that  the  maximum  tension  which 
accompanies  any  shearing  strain  in  such  a 
body  (good  iron  or  steel,  but  not  wood)  will 


=?J  .j^  —  a==-T  >  a  •  or  "  wto  maximum  ten 

sion  is  limited  to  10,000  Ibs.  per  sq.  in.,  no 
shearing  strain  must  be  greater  than  8,000 
Ibs.  per  sq.  in. 


33 


It  lias  been  shown  that  in  the  centre  of 
any  pin  the  shearing  strain  is  If  times 
greater  than  if  the  shearing  force  were  uni 
formly  distributed  over  the  cross  section, 
hence  the  pin  must  be  proportioned  to  with 
stand  a  uniformly  distributed  shearing  force 
of  If  times  the  actual  one  ;  in  other  words, 

the  shearing  strain  must  only  be—  1—  lbs.= 

IB 

5,810  Ibs.  per  sq.  in.* 

If  this  condition  is  observed  the  tension 
in  the  centre  of  the  pin  acting  at  45  deg.  to 
its  axis  will  be  not  more  than  10,000  Ibs. 
per  sq.  in. 

If  the  section  of  the  tie  bar  is  "5"  by  "$" 
the  total  shearing  force  will  be  10,000  X  & 

X  t  and  the  section  of  the  pin  will  be 


=  1.72  b  t.,  or  the  section  of  the  pin 
must  be  nearly  equal  1|  the  section  of  the 
bar. 

This  condition  determines  that  in  the 
neutral  axis  of  the  pin  the  shearing  strains, 
tensions,  and  compressions  shall  not  exceed 

*  The  Baltimore  Bridge  Company  make  the  uniformly  distrib 
uted  shearing  strain=6,000  Ibs.  per  sq.  in. 


34 


the  maximum  tension  usually  prescribed,  of 
10,000  Ibs.  per  sq.  in. 

The  rule  would  apply  if  another  amount, 
as  12,000  or  15,000,  were  prescribed.  The 
limit  of  shearing  strain  should  in  such  a 
case  be  raised  correspondingly,  by  still 
making  the  pin  section  If  times  the  section 
of  the  bar. 

It  will,  however,  be  seen  that  considera 
tion  of  the  shearing  strain  alone  is  not  suf 
ficient  to  properly  proportion  a  pin. 

The  results  obtained  thus  far  depend  on 
the  modulus,  G  being  f  of  E  for  a  homoge 
neous  body  as  good  iron  or  steel.  This  as 
sertion  must  now  be  proved.  By  purely 
mathematical  investigation,  Navier,  then 
Cauchy,  Dienger,  and  others  found  that 
any  pressure  on  a  perfectly  homogeneous 
body  is  accompanied  by  an  expansion  or 
lateral  tension  equal  to  J  of  the  pres 
sure  per  sq.  in.,  and  the  tension  accom 
panied  by  lateral  compression  equal  to 
J  the  value  of  the  tension.  Rude  exper 
iments  with  india-rubber  prove  the  exist 
ence  of  lateral  compression  or  tension,  and 
those  made  by  Wertheim  and  Begnauld 


35 


confirm  the  theory  sufficiently,  the  coeffi 
cient  differing  somewhat  with  the  degree  of 
homogeneity  of  the  bodies  under  test.  For 
iron,  Wertheim  found  J,  and  sometimes  a 
little  more,  but  not  so  much  as  to  cause  a 
change  in  the  modulus  to  exceed  1J  per 
cent.  Therefore,  without  entering  into  the 
analytical  investigations  of  Navier,  etc.,  it 
may  be  assumed  that  for  well-rolled  wrought 
iron  the  lateral  contraction  or  expansion  is 
J  the  longitudinal  tension  or  pressure. 


Tig.  6  represents  one  surface  of  a  hexahe- 


36 


dron,  which  of  infinitely  small  sides  is  as 
sumed  to  be  acted  upon  by  two  tensional 
forces  T,  equally  distributed  over  the  sur 
faces  indicated  in  the  figure  by  the  lines  AB 
and  oT>. 

The  sides  AD  .and  BC  are  extended  and 
the  sides  AB  and  oT)  compressed.  The  ex- 

T  T 

tension  will  be  —  BO.  the  compression  — 

AB,  or  since  AB=BC,  the  side  itself  can 
be  assumed  as  unit,  and  the  extension  will 

T 

be—-  whilst  the  lateral  contraction  is  J  of 
Is 

this  value. 

The  angle  BOO  was  originally  90  deg., 
but  now  it  is  more  and  the  increase  may  be 
represented  by  the  letter  e.  If  B  0  C  in 
crease,  the  angles  0  C  B  and  0  B  C  decrease 
each  one-half  the  amount  of  E. 

There  is  consequently  after  the  extension 

T 


tang 


A  B  after  the  contraction 
B  C  after  the  extension 

or  since  e  is  a  very  small  angle 


37 


••-  '•  -  -"-  tr\ 

tang  45°-  tang  -~       l-~        1~ 


1-ftang  45°tang~        1  +--•        1+  -r 
From  this  again  (because  e  is  very  small) 

0-£>0  -0=0-^)0  -I) 


or 

1  1          5T          A  5T- 

1  -*=1  —  T  =F;  and  s  =  -r  —  ; 

4:   E  4    Hi 

The  angle  A  0  B  having  changed  its  value, 
there  must  be  a  shearing  force  along  both 

diagonals. 

S          S 
This  shearing  force  being  St  «  will  be—  —and  — 


The  value  S  can  also  be  found  by  a  second 
consideration.  If  the  prism  A  B  D  is  in 
equilibrium,  the  force  T,  will  be  counteract 
ed  by  a  shearing  force  S,  and  a  tensional 
force  perpendicular  to  diagonal  B  D. 

The  projection  of  the  total  force  acting  on 
the  surface,  which  is  represented  by  the 
line  A  B,  on  the  diagonal  B  D,  must  equal 
the  total  shearing  force  acting  on  the  sur 
face  of  the  prism  represented  by  line  B  D, 
or 


38 


45°  =  H)XlxS,  or 

T  X  cos  45°=   1  X.?0.  or  T  =  2  S. 
cos  4:5° 

The  two  equations  for  S  combined  give  the 
value  of  G  represented  by  E,  so  that  there  is  : 

S         5      2  S  2  . 

or  G  =  —  E  - 


G         4        E  5 

This  is  the  equation  upon  which  the  rule 
was  based  that  the  section  of  a  pin  must  be 
at  least  If  times  the  section  of  the  bar,  to 
keep  the  tension  caused  by  the  shearing 
strain  below  the  limit  generally  prescribed. 
To  many  engineers  the  above  deductions 
may  be  new,  and  it  is  therefore  desirable  to 
dwell  for  a  few  moments  on  the  conclusions 
which  may  be  derived  therefrom.  It  has 
been  mentioned  that  "Wertheim  and  Eeg- 
nauld  made  several  series  of  experiments, 
by  which  they  established  that  the  factor  of 
lateral  contraction  does  most  sufficiently 
correspond  with  the  moduli  E  and  G,  as 
found  by  tensionaland  torsional  experiments. 

The  theory  of  strains  has  received  very 
valuable  proofs  by  Chief  Engineer  Woehler, 
in  Germany.  He  made  experiments  on 
impact,  on  tension,  compression,  torsion,  etc., 


39 


during  12  years.  He  first  established  the 
law  that  any  material  may  be  brSfe&t 
repeating  sufficiently  often  extensions, 
which,  however,  cause  strains  below  the 
breaking  point,  and  he  then  determined 
certain  limits  of  strain  within  which  the 
material  did  not  break.  Further  experi 
ments  on  torsion  and  other  more  direct 
chearing  strains  confirmed  the  same  law, 
but  established  that  this  limit  must  neces 
sarily  be  lower  then  for  direct  strains. 

We  may  call  these  limits  the  "  limits  of 
durability." 

Cast-steel,  cut  from  railway  axles  furnish 
ed  by  Krupp,  has  not  broken  after  40,000,- 
000  impacts,  straining  the  material  trans 
versely  to  53,000  Ibs.  Nor  could  this  steel 
be  broken  by  any  number  of  impacts  caus 
ing  shearing  strains  of  42,000  Ibs.  per 
square  inch.  All  strains  higher  than  these 
produced  rupture  after  a  sufficient  number 
of  impacts ;  and  there  seemed  to  exist  a  cer 
tain  relation  between  the  number  of  im 
pacts  and  the  value  of  the  strain.  We  may 
therefore  call  53,000  Ibs.  the  limit  of  dura 
bility  for  tensile,  and  42,000  Ibs.  the  limit 


40 


for  shearing  strain  of  the  class  of  steel  ex 
perimented  on. 

According  to  theory,  the  ratio  between 
both  limits  ought  to  be  as  1 :  0.800,  whilst  in 
reality  it  was  as  1  :  0.793. 

The  tensile  modulus  of  this  steel  was  ex 
perimented  on,  and  fixed  at  28,725,000  Ibs., 
while  the  shearing  modulus,  found  by  very 
careful  torsional  experiments,  was  11,237,- 
000  Ibs.  Their  ratio,  according  to  theory, 
ought  to  be  as  5 :  2. — when  the  result  of  the 
experiment  proved  it  to  be  as  5  :  1.95. 

The  quoted  results  refer  only  to  a  very 
small  part  of  the  experiments,  which  were 
extended  to  copper,  and  wrought  and  cast- 
iron  of  different  makes  ;  all  gave  results  in 
good  harmony  with  theory,  and  as  they 
were  made  to  test  the  material  in  the  same 
manner  as  in  practice,  they  prove  that, 
within  the  limits  for  which  the  theory 
holds  good,  we  can  well  rely  on  it  with 
this  proviso — that  for  riveted,  forged  or 
machine-worked  parts,  experiments  should 
be  made  of  each  class  of  material,  by  which 
alone  a  correct  idea  as  to  their  strength  and 
endurance  can  be  formed.  The  older  ex- 


41 


periments,  made  on  strains  beyond  the 
limits  of  durability  of  the  material,  or  not 
with  acting  in  the  same  manner  as  in  prac 
tice,  or  carried  to  the  ultimate  point  of 
failure,  could  not  possibly  lead  to  any  law 
or  reliable  formula,  because  the  phenomena 
beyond  the  elastic  limit  cannot  be  followed 
up  by  even  the  highest  analysis,  and  pos 
sibly  do  not  conform  to  any  law. 

Frequently,  as  a  consequence  of  the  older 
views,  the  effects  of  shearing,  torsional, 
compressive,  and  tensile  strains  and  strength, 
are  spoken  of  as  so  many  different  pheno 
mena  without  relation  to  each  other,  while 
in  reality  in  every  point  of  any  strained 
body  there  exist  shearing,  tensile,  and  com 
pressive  strains  at  the  same  time. 

Referring  to  the  object  of  this  paper,  it 
may  be  added  that  experiments  on  the 
ultimate  shearing  strength  show  that  the 
results  greatly  depend  on  the  way  they  are 
obtained,  in  some  cases  the  shearing 
strength  being  found  as  great  as  the  tensile 
strength,  and  in  other  cases  only  two-thirds 
of  it. 

If  the  greatest  tension  in  a  wrought-iron 


42 


bridge  is  assumed  at  10,000  Ibs.  per  square 
inch,  the  greatest  shearing  strain  ought  not 
to  exceed  8,000  Ibs.  This  is  far  within  the 
limit  of  durability  of  good  iron  under  ten 
sion,  which  after  132,000,000  of  impacts 
was  established  at  33,000  Ibs.  per  square 
inch. 

Having  now  determined  the  fundamental 
laws  of  elasticity  which  enter  into  the  prob 
lem  of  the  shearing  strains  in  a  pin,  and 
also  having  given  the  data  necessary  in 
estimating  the  maximum  pressure  that  can 
be  allowed  in  the  pin-hole,  a  more  impor 
tant  consideration  remains,  the  nature  of 
the  strain  caused  by  the  moment  of  flexure 
of  the  pin  is  still  to  be  examined. 

Attention  is  called  to  Figure  1,  represent 
ing  the  strains  upon  a  pin  either  in  a  top 
chord  of  a  truss  bridge,  at  the  exterior 
links  of  a  bottom  chord,  or  of  chain  suspen 
sion  bridges.  The  question  is  where  to 
find  the  maximum  moment  of  flexure  ? 

It  is  a  very  common  mistake  to  assume 
that  the  maximum  moment  is  in  section 
F  B,  when,  instead,  it  occurs  in  section 
H  C.  For  by  reversing  Fig.  1  it  will  be 


43 


seen  that  the  problem  is  the  same  as  if  a 
beam,  G  G,  were  loaded  on  both  sides  from 
G  to  H,  leaving  a  space  H  H  unloaded, 
and  that  the  two  eye-bars  form  the  supports 
of  such  a  beam. 

Consequently  the  moment  must  have  a 
constant  value  from  H  to  H,  and  the  maxi 
mum  moment  must  exist  in  any  section  of 
the  pin  between  H  and  H. 

The  pressure  on  B  C  for  the  present  pur 
pose  can  be  considered  as  uniformly  dis 
tributed,  and  the  strain  of  the  eye-bar 
assumed  as  being  concentrated  in  D,  which 
is  in  the  middle  of  line  M  G.  The  reac 
tion  likewise  is  then  concentrated  in  the 
middle  of  B  C,  which,  according  to  pre 
vious  considerations,  is  made  to  be  equal 
to  MG. 

In  fact  the  centre  of  the  reactionary 
forces  is  a  little  closer  to  B  than  to  0,  but 
the  difference  is  very  small,  as  long  as  the 
bearing  is  not  longer  than  the  bar  is  thick. 

This  supposition  leads  to  the  smallest 
possible  pin,  and  when  the  bearing  surface 
is  made  longer,  the  pin  must  be  stronger, 
while  the  pressure  in  the  pin  hole  at  the 


44 


same  time  decreases,  but  is  less  uniformly 
distributed. 

The  moment  of  flexure  of  the  pin  in  the 

section    HO,   is  =  P  (^^+  G  H^less 


f  TT 

P.  """—,  or  since  G  H  was  made  =  M  G 

A 

there  will  be  the  maximum  moment  = 
P  .  M  G  =  the  strain  of  the  eye-bar  multi 
plied  by  the  thickness  of  the  bar  as  lever. 

The  exact  value  of  the  moment  can  be  found  by 

d  v* 
determining  the  value  E  I       2  of  the  curve  B  C, 

(JL  £/ 

for  x  =  I  =  B  C  of  which  the  equation  has  been 
developed. 

This  value  for  any  length  P  of  B  C  would  give 
the  exact  moment  and  strain  which  a  pin  has  to 
bear,  its  diameter  being  assumed  previously 

_  M  G  +  G  H  . 

The  value  P  -  3  -  is  much  larger  than  it 

9 

would  have  been  obtained  had  section  F  B  been 
the  one  examined.  Then  the  moment  would  have 

.    _  M  G  GH 

been    only  P.  —  —  ,  which  is  less  by  P  .   —  —  — 
a  A 

than  the  actual  maximum  moment. 

Under  the  assumption  that  M  G  =  G  H 
the  error  is  100  per  cent,  and  when  G  H> 


45 


M  G  it  is  still  greater.  If  tlie  size  of  the  pin 
were  calculated  under  this  erroneous  sup 
position,  the  actual  strain  would  be  twice 
or  more  than  twice  as  large  as  was  intended. 
There  is  no  shearing  strain  and  no  compres 
sion  by  reaction  in  the  section  C  H  and 
the  tension  in  the  uppermost  fibres  could  be 
used  for  the  determination  of  the  size  of  pins. 

The  well  known  elements  of  flexure  prescribe  that 

S  /"  d  ~\* 

there  must  be,  for  I  =  t  :  -—  /  -—  J   .3.14  =P  t,  or 

since  P=&X  tX  10000  Ibs.  and  when  S  is  fixed  at 

10000  Ibs.  per  sq.  in.  ~p  3.14  =  5X  i*  and  d3  = 
10.18  .  b  I*. 

If  5  =4$  d=  /31/£O72  =3.44  1  =  0.86  width  of  bar. 
4  =3.18  1  =  1.06      "        " 


6=  t  d  =3-/10.18  =  2.16  t  =  2.16      " 

6=  t  (round  bar)  d==  t  8  V5"=2  .00  t  =2.00 
thickness  of  bar. 

These  sizes  of  pins  are  so  large  as  to  ex 
clude  entirely  the  consideration  of  shearing 
strain  and  of  pressure  in  the  pin-hole,  but 
it  may  be  justifiable  to  permit  greater  strains 
from  flexure  than  has  been  prescribed  for 
the  maximum  tension  of  the  tie-bar.  In- 


46 


deed  there  are  reasons  for  this  considera 
tion.  1st.  It  is  impossible  to  equally  strain 
all  bottom  chord  bars  to  exactly  the  speci 
fied  limit ;  some  bars  will  receive  tension 
exceeding  10,000  Ibs.  per  square  inch, 
as  will  be  explained  hereafter.  Also  when 
several  bars  are  fixed  to  the  same  pin  it 
cannot  be  expected  that  they  severally  will 
have  equal  moduli  of  elasticity,  hence  the 
prescribed  maximum  of  10,000  Ibs.  may  be 
exceeded. 

2d.  The  eye-bars  have  been  shaped  by 
a  process  of  manufacture,  which  under  all 
circumstances  somewhat  impairs  the  uni 
form  quality  of  the  iron. 

3d.  A  pin  has  not  been  exposed  to  fire 
after  having  been  rolled ;  if  turned  to  size 
with  good  machinery,  and  by  a  skilful 
workman,  its  quality  has  not  been  altered ; 
hence  the  iron  of  which  it  is  made  is  rnor.e 
reliable  than  that  of  any  other  part  of  the 
structure. 

Under  these  suppositions,  it  is  thought 
safe  to  permit  a  maximum  tension  of  12,000 
Ibs.  per  square  inch,  which  is  not  more  than 


47 


the  maximum  pressure  in  the  pin-hole  as 
determined  in  the  first  part  of  this  paper. 
The  results  of  the  calculation  brought  to 
practical  dimensions,  are  represented  in  the 
following  table : 

Width  of  bar b  4"  3"  2"  1"  1". 

Thickness  of  bar.  .t  1"  I"  1"  1  square  1  round. 

Diameter  of  pin. .  .d  3i"  3"  2J"  2"  1}" 

Batio  of  d  :  & 0.8,  1.0,  H  to  1|,  2,  If  to  If 

The  maximum  shearing  strain  of  these 
pins  is  6,600  Ibs.  per  square  inch — the 
minimum  about  5,000  Ibs. 

According  to  these  results  the  old  rule 

of  ~    =  -f  is  only  admissible  for  bars  with 

thickness  -J-  of  their  width,  or  less  than  this. 
The  use  of  this  rule  for  a  square  bar  would 
permit  a  maximum  strain  of  more  than 
60,000  Ibs.  per  square  inch,  if  the  condi 
tions  did  not  change  by  the  action  of  the 
nut  when  a  permanent  set  takes  place,  and 
a  pin  of  1£"  diameter  for  a  2"  X  1"  bar, 
would  permit  a  maximum  tension  of  nearly 
50,000  Ibs. 

These  two  examples  are  sufficient  to  show 
the  importance  of  properly  proportioned 


48 


pins,  and  that  the  best  American  bridge 
constructors  are  wise  in  using  dimensions 
varying  but  little  from  those  found  above.* 

Indeed  for  a  considerable  time  the  more 
scientific  engineers  of  this  country  have 
strengthened  by  degrees  their  bridge  con 
nections,  and  by  combining  practical  ex 
perience  with  theoretical  considerations  the 
theoretical  dimensions  have  been  nearly 
reached.  European  engineers  by  using 
riveted  work  have  not  had  an  opportunity 
to  obtain  this  result.  This  affords  a  reason 
why  in  many  late  books  the  subject  of  pins 
is  not  treated  more  scientifically. 

The  practical  bridge  builder,  when  de 
signing  a  structure,  sometimes  finds  it  diffi 
cult  to  fulfil  the  conditions  of  theory  and 
manufacture  which  more  or  less  contradict 
each  other.  However,  the  heavier  tie  bars 
which  carry  the  greater  part  of  the  dead 
load  are  not  so  much  influenced  by  the  live 
load,  which  is  the  really  destructive  element, 


*  Mr.  Thos.  C.  Clarke,  of  Philadelphia,  stated  in  a  recent 
letter  to  the  American  Society  of  Civil  Engineers,  that  in  the 
bridges  of  the  Phoenix  Bridge  Co.,  nearly  the  same  diameters 
as  given  in  the  last  table  are  used. 


49 


while  the  pins  of  the  lighter  bars,  near  the 
centre  of  the  top  chord,  are  generally 
stronger  than  necessary. 

The  diameter  of  top  chord  pins  can  be 
reduced  by  making  them  what  is  called 
double  shearing,  a  bearing  being  placed  on 
each  side  of  each  bar;  the  diameter  can 
then  be  determined  as  if  the  bar  had  only 
-j-%-  at  its  thickness. 

It  has  been  seen  how  necessary  it  is  to 
use  the  proper  size  of  pin  in  the  top  chords, 
which  is  more  especially  true  for  pins  bearing 
more  than  one  bar.  In  this  case  the  sum 
of  the  bending  moments  produced  by  the 
several  bars  must  be  introduced  in  the  for 
mula  for  the  tensile  strain  by  flexure. 

Great  care  is  also  required  in  properly 
proportioning  the  pins  of  bottom  chords  of 
bridges.  Generally  here  the  pin  takes  hold 
of  the  post,  directly  to  it  are  attached  the 
ties,  outside  of  which  the  bottom  bars  are 
arranged.  Here  the  greatest  moment  of  the 
pin  is  near  the  bottom  bar  next  to  the  tie. 
It  is  an  accumulation  arising  from  the  some 
times  great  number  of  bars.  In  order  to 
form  a  more  precise  idea  of  the  strains 


50 


which  may  arise,  it  may  be  as  well  to  ex 
amine  a  practical  example. 

Suppose  the  three  equally  strained  bot 
tom  bars  3  in-X^  in->  3  bars  8  in-XH  m-> 
and  a  tie  3  in-X^  m*  are  on  the  same  pin 
on  one  side  of  the  post  of  the  bridge.  The 
maximum  moment  will  be  5.1  in.X10>000 
Ibs.,  and  the  maximum  tensional  strain  by 
flexure  12,100  Ibs.,  whilst  the  pressure  in 
the  pin  hole  will  be  found  to  be  8,570  Ibs , 
resulting  in  a  tension  of  14,200  Ibs. 

Some  bridge-builders  use  square  bars, 
with  pins  not  much  larger  than  the  thick 
ness  of  the  bar. 

Suppose,  in  reference  to  the  above  ex 
ample,  that  three  2  in.  square  bottom  bars 
w^ere  counteracted  by  three  If  in.  square 
bars  and  the  tie.  For  a  3  in.  pin  the  ten 
sile  strain  by  flexure  would  amount  to  25,- 
000  Ibs.  Of  course  neither  this  nor  the 
above  strain  of  14,200  Ibs.  will  really  exist 
in  the  pin.  The  fact  will  be  that  the  pins 
bend  a  little  toward  the  centre  of  the  bridge, 
and  the  bars  near  to  the  centre  line  of  the 
bottom  chord  will  bear  a  greater  share  of 
the  chord  strain,  relieving  the  outer  bars. 


51 


The  difference  will  be  the  greater,  the 
smaller  the  pin  ;  the  greater  the  difference 
in  size  of  bars,  the  thicker  the  bars,  and  the 
shorter  will  be  the  panels. 

The  difficulty  of  reducing  the  strains  of 
bottom  pins  to  the  strain  of  the  bars  in 
creases  with  the  magnitude  of  the  bridge, 
and  may  partly  be  met  by  increasing  the 
number  of  flat  and  thin  bars,  making  up 
for  the  difference  in  tension  in  two  adjoin 
ing  panels  rather  by  the  number  than  by 
the  size  of  the  bars,  and  especially  by  using 
the  proper  size  of  pin. 

Another  way  to  meet  the  difficulty  con 
sists  in  creating  two  centre  lines  in  the  bot 
tom  chord,  by  placing  a  proper  number  of 
chord  bars  between  the  ties,  by  using  two 
posts,  or  by  constructing  a  properly  built 
post  foot.  For  small  bridges  these  costly 
arrangements  can  be  dispensed  with.  Flat 
bars,  in  preference  to  round  or  square  ones, 
are  in  all  cases  to  be  recommended.  The 
Baltimore  Bridge  Company,  for  instance, 
always  use  such,  and  it  seems  that  other 
constructors  have  recently  adopted  the  same, 
to  the  exclusion  of  round  bars. 


52 


Pins  used  in  chain  suspension  bridges, 
the  bars  being  all  of  equal  section,  have  to 
resist  only  a  small  moment ;  their  propor 
tions  follow  the  rule  for  top  pins. 

As  a  conclusion  to  which  the  foregoing 
examination  perhaps  has  led,  it  may  be 
mentioned  that  it  is  easier  to  calculate  the 
general  strains  of  skeleton  structures  than 
to  design  t  details,  which,  satisfying  the 
practical  demands  of  economical,  speedy,  and 
reliable  manufacture,  are  also  in  harmony 
with  the  more  subtle  ones  concerning  their 
proportions. 


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I21I10,  Cloth 2    OO 

THE  EARTH'S  CRUST.  A  handy  Outline  of  Geo 
logy.  By  David  Page.  Illustrated,  i8mo,  cloth. ...  75 

DICTIONARY  of  Manufactures,  Mining,  Machinery, 
and  the  Industrial  Arts.  By  George  Dodd.  i2mo, 
cloth 2  oc 

FRANCIS.  On  the  Strength  of  Cast-Iron  Pillars,  with 
Tables  for  the  use  of  Engineers,  Architects,  and 
Builders.  By  James  B.  Francis,  Civil  Engineer, 
i  vol.  8vo,  cloth 2  OG 


D.   YAN   NOSTBAND  S   PUBLICATIONS. 

GILLMORE  (Gen.  Q.  A.)  Treatise  on  Limes,  Hy 
draulic  Cements,  and  Mortars.  Papers  on  Practical 
Engineering,  U.  S.  Engineer  Department,  No.  9, 
containing  Reports  of  numerous  Experiments  con 
ducted  in  New  York  City,  during  the  years  1858  to 
1861,  inclusive.  By  Q.  A.  Gillmore,  Bvt.  Maj  -Gen., 
U.  S.  A.,  Major,  Corps  of  Engineers.  With  num 
erous  illustrations,  i  vol,  8vo,  cloth $4  oo 

HARRISON.  The  Mechanic's  Tool  Book,  with  Prac 
tical  Rules  and  Suggestions  for  Use  of  Machinists, 
Iron  Workers,  and  others.  By  W.  B.  Harrison, 
associate  editor  of  the  "American  Artisan."  Illus 
trated  with  44  engravings,  izmo,  cloth i  50 

HENRICI  (Olaus).  Skeleton  Structures,  especially  in 
their  application  to  the  Building  of  Steel  and  Iron 
Bridges.  By  Olaus  Henrici.  With  folding  plates 
and  diagrams,  i  vol.  8vo,  cloth 3  oo 

HEWSON  (Win.)  Principles  and  Practice  of  Embank 
ing  Lands  from  River  Floods,  as  applied  to  the  Le 
vees  of  the  Mississippi.  By  William  Hewson,  Civil 
Engineer,  i  vol.  Svo,  cloth 2  oo 

HOLLEY  (A.  L.)  Railway  Practice.  American  and 
European  Railway  Practice,  in  the  economical  Gen 
eration  of  Steam,  including  the  Materials  and  Con 
struction  of  Coal-burning  Boilers,  Combustion,  the 
Variable  Blast,  Vaporization,  Circulation,  Superheat 
ing,  Supplying  and  Heating  Feed-water,  etc.,  and 
the  Adaptation  of  Wood  and  Coke-burning  Engines 
to  Coal-burning ;  and  in  Permanent  Way,  including 
Road-bed,  Sleepers,  Rails,  Joint-fastenings,  Street 
Railways,  etc.,  etc.  By  Alexander  L.  Holley,  B.  P. 
With  77  lithographed  plates,  i  vol.  folio,  cloth.  ...  12  oo 

KING  (W.  H.)  Lessons  and  Practical  Notes  on  Steam, 
the  Steam  Engine,  Propellers,  etc.,  etc.,  for  Young 
Marine  Engineers,  Students,  and  others.  By  the 
late  W.  H.  King,  U.  S.  Navy.  Revised  by  Chief 
Engineer  J.  W.  King,  U.  S.  Navy.  Twelfth  edition, 
enlarged.  Svo,  cloth 2  oo 

MINIFIE  (Wra.)  Mechanical  Drawing.  A  Text-Book 
of  Geometrical  Drawing  for  the  use  of  Mechanic* 

4 


£.   VAN   NOSTBAND  S  PUBLICATIONS. 

Schools,  in  which  the  Definitions  and  Rules  of 
Geometry  are  familiarly  explained;  the  Practical 
Problems  are  arranged,  from  the  most  simple  to  the 
more  complex,  and  in  their  description  technicalities 
are  avoided  as  much  as  possible.  With  illustrations 
for  Drawing  Plans,  Sections,  and  Elevations  of  Rail 
ways  and  Machinery ;  ah  Introduction  to  Isometrical 
Drawing,  and  an  Essay  on  Linear  Perspective  and 
Shadows.  Illustrated  with  over  200  diagrams  en 
graved  on  steel.  By  Wm.  Minifie,  Architect.  Sev 
enth  edition.  With  an  Appendix  on  the  Theory  and 
Application  of  Colors,  i  vol.  8vo,  cloth $4  oo 

"It  Is  the  best  work  on  Drawing  that  we  have  ever  seen,  and  is 
especially  a  text-book  of  Geometrical  Drawing  lor  the  use  of  Mechanic* 
and  Schools.  No  young  Mechanic,  such  as  n,  Machinists,  Engineer,  Cabi 
net-maker,  Millwright,  or  Carpenter,  should  be  without  it."— Scientific 
American. 

.  .  Geometrical  Drawing.  Abridged  from  the  octavo 
edition,  for  the  use  of  Schools.  Illustrated  with  48 
steel  plates.  Fifth  edition,  i  vol.  i2mo,  cloth....  2  oc 

STILLMAN  (Paul.)  Steam  Engine  Indicator,  and  the 
Improved  Manometer  Steam  and  Vacuum  Gauges — 
their  Utility  and  Application.  By  Paul  Stillman. 
New  edition,  i  vol.  ismo,  flexible  cloth x  oo 

SWEET  (S.H.)  Special  Report  on  Coal;  showing  its 
Distribution,  Classification,  and  cost  delivered  over 
different  routes  to  various  points  in  the  State  of  New 
York,  and  the  principal  cities  on  the  Atlantic  Coast. 
By  S.  H.  Sweet.  With  maps,  i  vol.  8vo,  cloth 3  oo 

WALKER  (W.  H.)  Screw  Propulsion.  Notes  on 
Screw  Propulsion  :  its  Ris«  and  History.  By  Capt. 
W.  H.  Walker,  U,  S.  Navy,  i  vol.  8vo,  cloth 75 

WARD  (J.  H.)  Steam  for  the  Million.  A  popular 
Treatise  on  Steam  and  its  Application  to  the  Useful 
Arts,  especially  to  Navigation.  By  J.  H.  Wand, 
Commander  U.  S.  Navy.  New  and  revised  edition. 
z  vol.  8vo,  cloth z  oo 

WEISBACH  (Julius).  Principles  of  the  Mechanics  of 
Machinery  and  Engineering.  By  Dr.  Julius  Weis- 
bach,  of  Freiburg.  Translated  from  the  last  German 
edition,  i  Vol.  I.,  8 vo,  cloth 10  oo 


D.  VAN  NOSTBAND'S  PUBLICATIONS. 

DIEDRICH.  The  Theory  of  Strains,  a  Compendium 
for  the  calculation  and  construction  of  Bridges,  Roofs, 
and  Cranes,  with  the  application  of  Trigonometrical 
Notes,  containing  the  most  comprehensive  informa 
tion  in  regard  to  the  Resulting  strains  for  a  perman 
ent  Load,  as  also  for  a  combined  (Permanent  and 
Rolling)  Load.  In  two  sections,  adadted  to  the  re 
quirements  of  the  present  time.  By  John  Diedrich, 
0.  E.  Illustrated  by  numerous  plates  and  diagrams, 
8vo,  cloth ,,.....,.,....... 5  oo 

WILLIAMSON  (R.  S.)  On  the  use  of  the  Barometer  on 
Surveys  and  Reconnoissances.  Part  I.  Meteorology 
in  its  Connection  with  Hypsometry.  Part  II.  Baro 
metric  Hypsometry.  By  R.  S.  Wiliamson,  Bvt. 
Lieut. -Col.  U.  S.  A.,  Major  Corps  of  Engineers. 
With  Illustrative  Tables  and  Engravings.  Paper 
No.  15,  Professional  Papers,  Corps  of  Engineers, 
i  vol.  4to,  cloth 15  oo 

POOK  (S.  M.)  Method  of  Comparing  the  Lines  and 
Draughting  Vessels  Propelled  by  Sail  or  Steam. 
Including  a  chapter  on  Laying  off  on  the  Mould- 
Loft  Floor.  By  Samuel  M.  Pook,  Naval  Construc 
tor,  i  vol.  Svo,  with  illustrations,  cloth 5  oo 

ALEXANDER  (J.  H.)  Universal  Dictionary  of 
Weights  and  Measures,  Ancient  and  Modern,  re 
duced  to  the  standards  of  the  United  States  of  Ame 
rica.  By  J.  H.  Alexander.  New  edition,  enlarged, 
i  vol.  Svo,  cloth 3  50 

BROOKLYN  WATER  WORKS.  Containing  a  De 
scriptive  Account  of  the  Construction  of  the  Works, 
and  also  Reports  on  the  Brooklyn,  Hartford,  Belle 
ville  and  Cambridge  Pumping  Engines.  With  illustra 
tions,  i  vol.  folio,  cloth 

RICHARDS'  INDICATOR.  A  Treatise  on  the  Rich 
ards  Steam  Engine  Indicator,  with  an  Appendix  by 
F.  W.  Bacon,  M.  E.  i8mo,  flexible,  cloth i  oo 


IX  VAN  NOSTKAND  S  PUBLICATIONS. 


POPE.  Modern  Practice  of  the  Electric  Telegraph.  A 
Hand  Book  for  Electricians  and  operators.  By  Frank 
L.  Pope.  Eighth  edition,  revised  and  enlarged,  and 
fully  illustrated.  8vo,  cloth $2.00 

"  There  is  no  other  work  of  this  kind  in  the  English  language  that  con 
tains  in  so  small  a  compass  so  much  practical  information  in  the  appli 
cation  of  galvanic  electricity  to  telegraphy.  It  should  be  in  the  hands  of 
every  one  interested  in.  telegraphy,  or  the  use  of  Batteries  for  other  pur 
poses." 

MORSE.  Examination  of  the  Telegraphic  Apparatus 
and  the  Processes  in  Telegraphy.  By  Samuel  F. 
Morse,  LL.D.,  U.  S.  Commissioner  Paris  Universal 
Exposition,  1867.  Illustrated,  8vo,  cloth $2  oo 

SABINE.  History  and  Progress  of  the  Electric  Tele 
graph,  with  descriptions  of  some  of  the  apparatus. 
By  Robert  Sabine,  C.  E.  Second  edition,  with  ad 
ditions,  izmo,  cloth i  25 

CULLEY.  A  Hand-Book  of  Practical  Telegraphy.  By 
R.  S.  Culley,  Engineer  to  the  Electric  and  Interna 
tional  Telegraph  Company.  Fourth  edition,  revised 
and  enlarged.  Illustrated  8 vo,  cloth 500 

BENET.  Electro-Ballistic  Machines,  and  the  Schultz 
Chronoscope.  By  Lieut. -Col.  S.  V.  Benet,  Captain 
of  Ordnance,  U.  S.  Army.  Illustrated,  second  edi 
tion,  4to,  cloth 3  oo 

MICHAELIS.  The  Le  Boulenge  Chronograph,  with 
three  Lithograph  folding  plates  of  illustrations.  By 
Brevet  Captain  O.  E.  Michaelis,  First  Lieutenant 
Ordnance  Corps,  U.  S .  Army,  410,  cloth 3  oo 

ENGINEERING  FACTS  AND  FIGURES  An 
Annual  Register  of  Progress  in  Mechanical  Engineer 
ing  and  Construction,  for  the  years  1863,  64,  65,  66, 
67,  68.  Fully  illustrated,  6  vols.  i8mo,  cloth,  $2.50 
per  vol.,  each  volume  sold  separately 

HAMILTON^  Useful  Information  for  Railway  Men. 
Compiled  by  W.  Gi  Hamilton,  Engineer.  Fifth  edi 
tion,  revised  and  enlarged,  562  pages  Pocket  form. 

Morocco,  gilt... 2  oo 

7 


3).  VAN  NOSTRAND  S  PUBLICATIONS. 

STUART.  The  Civil  and  Military  Engineers  of  Amer 
ica.  By  Gen.  C.  B.  Stuart.  With  9  finely  executed 
portraits  of  eminent  engineers,  and  illustrated  by 
engravings  of  some  of  the  most  important  works  con 
structed  in  America.  8vo,  cloth $5  oo 

CTONEY.  The  Theory  of  Strains  in  Girders  and  simi 
lar  structures,  with  observations  on  the  application  of 
Theory  to  Practice,  and  Tables  of  Strength  and  other 
properties  of  Materials.  By  Bindon  B.  Stoney,  B.  A. 
New  and  revised  edition,  enlarged,  with  numerous 
engravings  on  wood,  by  Oldham.  Royal  8vo,  664 
pages.  Complete  in  one  volume.  8vo,  cloth 1500 

SHREVE.  A  Treatise  on  the  Strength  of  Bridges  and 
Roofs.  Comprising  the  determination  of  Algebraic 
formulas  for  strains  in  Horizontal,  Inclined  or  Rafter. 
Triangular,  Bowstring,  Lenticular  and  other  Trusses, 
from  fixed  and  moving  loads,  with  practical  applica 
tions  and  examples,  for  the  use  of  Students  and  Engi 
neers.  By  Samuel  H.  Shreve,  A.  M. ,  Civil  Engineer. 
87  wood  cut  illustrations.  8vo,  cloth 5  oo 

MERRILL.  Iron  Truss  Bridges  for  Railroads.  The 
method  of  calculating  strains  in  Trusses,  with  a  care 
ful  comparison  of  the  most  prominent  Trusses,  in 
reference  to  economy  in  combination,  etc.,  etc.  By 
Brevet.  Col.  William  E.  Merrill,  U  S.  A.,  Major 
Corps  of  Engineers,  with  nine  lithographed  plates  of 
Illustrations.  410,  cloth 500 

WHIPPLE.  An  Elementary  and  Practical  Treatise  on 
Bridge  Building.  An  enlarged  and  improved  edition 
of  the  author's  original  work.  By  S.  Whipple,  C.  E-, 
inventor  of  the  Whipple  Bridges,  &c.  Illustrated 
8vo,  cloth 4  oo 

THE  KANSAS  CITY  BRIDGE.  With  an  account 
of  the  Regimen  of  the  Missouri  River,  and  a  descrip 
tion  of  the  methods  used  for  Founding  in  that  River. 
By  O.  Chanute,  Chief  Engineer,  and  George  Morri 
son,  Assistant  Engineer.  Illustrated  with  five  litho 
graphic  views  and  twelve  plates  of  plans.  4to,  cloth,  6  oo 

8 


D.  Y^:T  KOSTRAND'S  PUBLICATIONS. 


MAC  CORD.  A  Practical  Treatise  on  the  Slide  Valve 
by  Eccentrics,  examining  by  methods  the  action  of  the 
Eccentric  upon  the  Slide  Valve,  and  explaining  the 
Practical  processes  of  laying  out  the  movements, 
adapting  the  valve  for  its  various  duties  in  the  steam 
engine.  For  the  use  of  Engineers,  Draughtsmen, 
Machinists,  and  Students  of  Valve  Motions  in  gene 
ral.  By  C.  W.  Mac  Cord,  A.  M.,  Professor  of  Me 
chanical  Drawing,  Stevens'  Institute  of  Technology, 
Hoboken,  N.  J.  Illustrated  by  8  full  page  copper 
plates.  410, cloth.; $4  °° 

KIRKWOOD.  Report  on  the  Filtration  of  River 
Waters,  for  the  supply  of  cities,  as  practised  in 
Europe,  made  to  the  Board  of  Water  Commissioners 
of  the  City  of  St.  Louis.  By  James  P.  Kirkwood. 
Illustrated  by  30  double  plate  engravings.  4to,  cloth,  15  oo 

PLATTNER.  Manual  of  Qualitative  and  Quantitative 
Analysis  with  the  Blow  Pipe.  From  the  last  German 
edition,  revised  and  enlarged.  By  Prof.  Th.  Richter, 
of  the  Royal  Saxon  Mining  Academy.  Translated 
by  Prof.  H.  B.  Cornwall,  Assistant  in  the  Columbia 
School  of  Mines,  New  York,  assisted  by  John  H. 
Caswell.  Illustrated  with  87  wood  cuts,  and  one 
lithographic  plate.  Second  edition,  revised,  560 
pages,  8vo,  cloth 7  50 

PLYMPTON.  The  Blow  Pipe.  A  system  of  Instruc 
tion  in  its  practical  use  being  a  graduated  course  of 
analysis  for  the  use  of  students,  and  all  those  engaged 
in  the  examination  of  metallic  combinations.  Second 
edition,  with  an  appendix  and  a  copious  index.  By 
Prof.  Geo  W.  Plympton,  of  the  Polytechnic  Insti 
tute,  Brooklyn,  N.  Y.  i2mo,  cloth 2  oo 

PYNCHON.  Introduction  to  Chemical  Physics,  design 
ed  for  the  use  of  Academies,  Colleges  and  High 
Schools.  Illustrated  with  numerous  engravings,  and 
containing  copious  experiments  with  directions  for 
preparing  them.  By  Thomas  Ruggles  Pynchon, 
M.  A.,  Professor  of  Chemistry  and  tne  Natural  Sci 
ences,  Trinity  College,  Hartford  New  edition,  re 
vised  and  enlarged,  and  illustrated  by  269  illustrations 
on  wood.  Crown,  Svo.  cloth 3  oo 

9 


r>.  VAN  NOSTRAND'S  PUBLICATIONS. 

ELIOT  AND  STORER.  A  C9mpendious  Manual  of 
Qualitative  Chemical  Analysis.  By  Charles  W. 
Eliot  and  Frank  H.  Storer.  Revised  with  the  Co 
operation  of  the  authors.  By  William  R.  Nichols, 
Professor  of  Chemistry  in  the  Massachusetts  Insti 
tute  of  Technology.  Illustrated,  lamo,  cloth $i  50 

RAMMELSBERG.  Guide  to  a  course  of  Quantitative 
Chemical  Analysis,  especially  of  Minerals  and  Fur 
nace  Products.  Illustrated  by  Examples-  By  C.  F. 
Rammelsberg.  Translated  by  J.  Towler,  M.  D. 
8vo,  cloth 2  25 

EGLESTON.  Lectures  on  Descriptive  Mineralogy,  de 
livered  at  the  School  of  Mines,  Columbia  College. 
By  Professor  T.  Egleston.  Illustrated  by  34  Litho 
graphic  Plates.  8vo,  cloth 4  50 

MITCHELL.  A  Manual  of  Practical  Assaying.  By 
John  Mitchell.  Third  edition.  Edited  by  William 
Crookes,  F.  R.  S.  8vo,  cloth 10  oo 

WATT'S  Dictionary  of  Chemistry.  New  and  Revised 
edition  complete  in  6  vols.  8vo  cloth,  $62.00.  Sup 
plementary  volume  sold  separately.  Price,  cloth. .  -  9  oo 

RANDALL.  Quartz  Operators  Hand-Book.  By  P.  M. 
Randall.  New  edition,  revised  and  enlarged,  fully 
illustrated,  i  zmo,  cloth 2  oo 

SILVERSMITH.  A  Practical  Hand-Book  for  Miners, 
Metallurgists,  and  Assayers,  comprising  the  most  re 
cent  improvements  in  the  disintegration,  amalgama 
tion,  smelting,  and  parting  of  the  Precious  ores,  with 
a  comprehensive  Digest  of  the  Mining  Laws.  Greatly 
augmented,  revised  and  corrected.  By  Julius  Silver-  , 
smith.  Fourth  edition.  Profusely  illustrated.  i2ino, 
cloth 3  oo 

THE  USEFUL  METALS  AND  THEIR  ALLOYS, 

including  Mining  Ventilation,  Mining  Jurisprudence, 
and  Metallurgic  Chemistry  employed  in  the  conver 
sion  of  Iron,  Copper,  Tin,  Zinc,  Antimony  and  Lead 
ores,  with  their  applications  to  the  Industrial  Arts. 
By  Scoffren,  Truan,  Clay,  Oxland,  Fairbairn,  and 

ethers.     Fifth  edition,  half  calf , 3  75 

10 


D.  VAN  NOSTRAND  S  PUBLICATIONS. 

JOYNSON.  The  Metals  used  in  construction,  Iron, 
Steel,  Bessemer  Metal,  etc.,  etc.  By  F.  H.  Joynson. 
Illustrated,  xamo,  cloth $o  75 

VON  COTTA.  Treatise  on  Ore  Deposits.  By  Bern- 
hard  Von  Cotta,  Professor  of  Geology  in  the  Royal 
School  of  Mines,  Freidberg,  Saxony.  Translated 
from  the  second  German  edition,  by  Frederick 
Prime,  Jr.,  Mining  Engineer,  and  revised  by  the  au 
thor,  with  numerous  illustrations.  8vo,  cloth 4  oo 

URE.  Dictionary  of  Arts,  Manufactures  and  Mines.  By 
Andrew  Ure,  M.D.  Sixth  edition,  edited  by  Robert 
Hunt,  F.  R..  S.,  greatly  enlarged  and  re-written. 
London,  1872.  3  vols.  8vo,  cloth,  $25.00.  Half 
Russia 37  50 

BELL.  Chemical  Phenomena  of  Iron  Smelting.  An 
experimental  and  practical  examination  of  the  cir 
cumstances  which  determine  the  capacity  of  the  Blast 
Furnace,  The  Temperature  of  the  air,  and  the 
proper  condition  of  the  Materials  to  be  operated 
upon.  By  I.  Lowthian  Bell.  8 vo,  cloth 600 

ROGERS.  The  Geology  of  Pennsylvania.  A  Govern 
ment  survey,  with  a  general  view  of  the  Geology  of 
the  United  States,  Essays  on  the  Coal  Formation  and 
its  Fossils,  and  a  description  of  the  Coal  Fields  of 
North  America  and  Great  Britain.  By  Henry  Dar 
win  Rogers,  late  State  Geologist  of  Pennsylvania, 
Splendidly  illustrated  with  Plates  and  Engravings  in 
the  text.  3  vols.,  4to,  cloth,  with  Portfolio  of  Maps.  30  oo 

BURGH.  Modern  Marine  Engineering,  _  applied  to 
Paddle  and  Screw  Propulsion.  Consisting  of  36 
colored  plates,  259  Practical  Wood  Cut  Illustrations, 
and  403  pages  of  descriptive  matter,  the  whole  being 
an  exposition  of  the  present  practice  of  James 
Watt  &  Co.,  J.  &  G.  Rennie,  R.  Napier  &  Sons, 
and  other  celebrated  firms,  by  N.  P.  Burgh,  Engi 
neer,  thick  410,  vol.,  cloth,  $25.00 ;  half  mor 30  oo 

BARTOL.   Treatise  on  the  Marine  Boilers  of  the  United 

States.    By  B.  H.  Bartol.    Illustrated,  8vo,  cloth.. .     150 

II 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 

BOURNE.  Treatise  on  the  Steam  Engine  in  its  various 
applications  to  Mines,  Mills,  Steam  Navigation, 
Railways,  and  Agriculture,  with  the  theoretical  in 
vestigations  respecting  the  Motive  Power  of  Heat, 
and  the  proper  proportions  of  steam  engines.  Elabo 
rate  tables  of  the  right  dimensions  of  every  part,  and 
Practical  Instructions  for  the  manufacture  and  man 
agement  of  every  species  of  Engine  in  actual  use. 
By  John  Bourne,  being  the  ninth  edition  of  "  A 
Treatise  on  the  Steam  Engine,"  by  the  "Artizan 
Club."  Illustrated  by  38  plates  and  546  wood  cuts. 
4to,  cloth $15  oo 

STUART.  The  Naval  Dry  Docks  of  the  United 
Slates.  By  Charles  B.  Stuart  late  Engineer-in-Chief 
of  the  U.  S.  Navy.  Illustrated  with  24  engravings 
on  steel.  Fourth  edition,  cloth 6  oo 

EADS.     System  of  Naval  Defences.      By  James  B. 

Eads,  0.  E.,  with  10  illustrations,  4to,  cloth 5  oo 

FOSTER.  Submarine  Blasting  in  Boston  Harbor. 
Massachusetts.  Removal  of  Tower  and  (Jorwin 
Rocks.  By  J.  G.  Foster,  Lieut -Col.  of  Engineers, 
U.  S.  Army.  Illustrated  with  seven  plates,  4to, 
cloth 3  50 

BARNES  Submarine  Warfare,  offensive  and  defensive, 
including  a  discussion  of  the  offensive  Torpedo  Sys 
tem,  its  effects  upon  Iron  Clad  Ship  Systems  and  in 
fluence  upon  future  naval  wars.  By  Lieut. -Com 
mander  J.  S.  Barnes,  U.  S.  NM  with  twenty  litho 
graphic  plates  and  many  wood  cuts,  8vo,  cloth 5  oo 

HOLLEY.  A  Treatise  on  Ordnance  and  Armor,  em 
bracing  descriptions,  discussions,  and  professional 
opinions  concerning  the  materials,  fabrication,  re 
quirements,  capabilities,  and  endurance  of  European 
and  American  Guns,  for  Naval,  Sea  Coast,  and  Iron 
Clad  Warfare,  and  their  Rifling,  Projectiles,  and 
Breech-Loading;  also,  results  of  experiments  against 
armor,  from  official  records,  with  an  appendix  refer 
ring  to  Gun  Cotton,  Hooped  Guns,  etc.,  etc.  By 
Alexander  L.  Holley,  B.  P.,  948  pages,  493  engrav 
ings,  and  747  Tables  of  Results,  etc.,  8vo,  half  roan,  xo  oo 

12 


D.  VAN  NOSTRAND'S  PUBLICATIONS. 

SIMMS.  A  Treatise  on  the  Principles  and  Practice  of 
Levelling,  showing  its  application  to  purposes  of 
Railway  Engineering  and  the  Construction  of  Roads, 
&c.  By  Frederick  W.  Simms,  C.  E.  From  the  sth 
London  edition,  revised  and  corrected,  with  the  addi 
tion  of  Mr.  Laws's  Practical  Examples  for  setting 
out  Railway  Curves.  Illustrated  with  three  Litho 
graphic  plates  and  numerous  wood  cuts.  8vo,  cloth.  $2  50 

BURT.  Key  to  the  Solar  Compass,  and  Surveyor's 
Companion  ;  comprising  all  the  rules  necessary  for 
use  in  the  field ;  also  description  of  the  Linear  Sur 
reys  and  Public  Land  System  of  the  United  States, 
Notes  on  the  Barometer,  suggestions  for  an  outfit  for 
a  survey  of  four  months,  etc.  By  W.  A.  Hurt,  U.  S. 
Deputy  Surveyor.  Second  edition.  Pocket  book 
form,  tuck, 2  50 

THE  PLANE  TABLE.  Its  uses  in  Topographical 
Surveying,  from  the  Papers  of  the  U.  S.  Coast  Sur 
vey.  Illustrated,  8vo,  cloth 200 

"This  worK  gives  a   description  of  the  Plane  Table,  employed  at  tha 
U.  S.  Coast  survey  office,  and  the  manner  of  using  it." 

JEFFER'S.  Nautical  Surveying:  By  W.  N.  Jeffers, 
Captain  U.  S.  Navy.  Illustrated  with  9  copperplates 
and  31  wood  cut  illustrations.  8vo,  cloth 5  oo 

CHAUVENET.  New  method  of  correcting  Lunar  Dis 
tances,  and  improved  method  of  Finding  the  error 
and  rate  of  a  chronometer,  by  equal  altitudes.  By 
W.  Chauvenet,  LL,  D.  8vo,  cloth 2  oo 

BRUNNOW.  Spherical  Astronomy.  By  F.  Brunnow, 
Hh.  Dr.  Translated  by  the  author  from  the  second 
German  edition.  8vo,  cloth 6  50 

PEIRCE.     System   of  Analytic.  Mechanics.     By  Ben- 

j  amin  Peirce.     4to,  cloth 10  oo 

COFFIN.  Navigation  and  Nautical  Astronomy.  Pre 
pared  for  the  use  of  the  U.  S.  Naval  Academy.  By 
Prof.  J.  H.  C.  Coffin.  Fifth  edition.  52  wood  cut  illus 
trations.  i2mo,  cloth 3  50 

13 


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CLARK.  Theoretical  Navigation  and  Nautical  Astron 
omy.  By  Lieut.  Lewis  Clark,  U.  S.  N.  Illustrated 
with  41  wood  cuts.  8vo,  cloth $3  oo 

HASKINS.  The  Galvanometer  and  its  Uses.  A  Man 
ual  for  Electricians  and  Students.  By  C.  H.  Has- 
kins.  1 21110,  pocket  form,  morocco.  (In  press) 

GOUGE.     New  System  of  Ventilation,  which  has  been 
thoroughly  tested,  under  the  patronage  of  many  dis 
tinguished  persons.    By   Henry  A.   Gouge.    With     » 
many  illustrations.    8vo,  cloth 200 

BECKWITH.  Observations  on  the  Materials  and 
Manufacture  of  Terra-Cotta,  Stone  Ware,  Fire  Brick, 
Porcelain  and  Encaustic  Tiles,  with  remarks  on  the 
products  exhibited  at  the  London  International  Exhi 
bition,  1871.  By  Arthur  Beckwith,  C.  E.  8vo, 
paper 60 

MORFIT.  A  Practical  Treatise  on  Pure  Fertilizers,  and 
the  chemical  conversipn  of  Rock  Guano,  Marlstones, 
Coprolites,  and  the  Crude  Phosphates  of  Lime  and 
Alumina  generally,  into  various  valuable  products. 
By  Campbell  Mornt,  M.D.,  with  28  illustrative  plates, 
8vo,  cloth 20  oa 

BARNARD.  The  Metric  System  of  Weights  and 
Measures.  An  address  delivered  before  the  convoca 
tion  of  the  University  of  the  State  of  New  York,  at 
Albany,  August,  1871.  By  F.  A.  P.  Barnard,  LL.D., 
President  of  Columbia  College,  New  York.  Second 
edition  from  the  revised  edition,  printed  for  the  Trus 
tees  of  Columbia  College.  Tinted  paper,  8vo,  cloth  3  oo 

.  Report  on  Machinery  and  Processes  on  the  In 
dustrial  Arts  and  Apparatus  of  the  Exact  Sciences. 
By  F.  A.  P.  Barnard,  LL.D.  Paris  Universal  Ex 
position,  1867.  Illustrated,  8vo,  cloth 5  oo 

BARLOW.  Tables  of  Squares,  Cubes,  Square  Roots, 
Cube  Roots,  Reciprocals  of  all  integer  numbers  up  frc 
10,000.  New  edition,  i2mo,  cloth 250 

14 


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MYER.  Manual  of  Signals,  for  the  use  of  Signal  officers 
in  the  Field,  and  for  Military  and  Naval  Students, 
Military  Schools,  etc.  A  new  edition  enlarged  and 
illustrated.  By  Brig.  General  Albert  J.  Myer,  Chief 
Signal  Officer  of  the  army,  Colonel  of  the  Signal 
Corps  during  the  War  of  the  Rebellion.  lamo,  48 
plates,  full  Roan • $5  oo 

WILLIAMSON.  Practical  Tables  in  Meteorology  and 
Hypsometry,  in  connection  with  the  use  of  the  Bar 
ometer.  By  CoL  R.  S.  Williamson,  U.  S-  A.  410, 
cloth 2  50 

THE  YOUNG  MECHANIC.  Containing  directions 
for  the  use  of  all  kinds  of  tools,  and  for  the  construc 
tion  of  Steam  Engines  and  Mechanical  Models,  in 
cluding  the  Art  of  Turning  in  Wood  and  Metal.  By 
the  author  "  The  Lathe  and  its  Uses,"  etc.  From 
the  English  edition  with  corrections.  Illustrated, 
i2mo,  cloth i  75 

PICKERT  AND  METCALF.  The  Art  of  Graining. 
How  Acquired  and  How  Produced,  with  description 
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and  Abraham  Metcalf.  Beautifully  illustrated  with 
42  tinted  plates  of  the  various  woods  used  in  interior 
finishing.  Tinted  paper,  4to,  cloth 10  oo 

HUNT.  Designs  for  the  Gateways  of  the  Southern  En 
trances  to  the  Central  Park.  By  Richard  M.  Hunt. 
With  a  description  of  the  designs.  4to.  cloth 500 

LAZELLE.  One  Law  in  Nature.  By  Capt.  H.  M. 
Lazelle,  U.  S.  A.  A  new  Corpuscular  Theory,  com 
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its  Multiple  Atom  Constitution,  applied  to  the  Physi 
cal  Affections  or  Modes  of  Energy.  i2mo,  cloth. . .  i  50 

PETERS.  Notes  on  the  Origin,  Nature,  Prevention, 
and  Treatment  of  Asiatic  Cholera.  By  John  C. 
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I2IHO,   Cloth I   SO 

15 


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BOYNTON.  History  of  West  Point,  its  Military  Im 
portance  during  the  American  Revolution,  and  the 
Origin  and  History  of  the  U.  S.  Military  Academy. 
By  Bvt  Major  C.  E.  Boynton,  A.M.,  Adjutant  of  the 
Military  Academy.  Second  edition,  416  pp.  8yo, 
printed  on  tinted  paper,  beautifully  illustrated  with 
36  maps  and  fine  engravings,  chiefly  from  photo 
graphs  taken  on  the  spot  by  the  author.  Extra 
cloth $3  50 

WOOD.  West  Point  Scrap  Book,  being  a  collection  of 
Legends,  Stories,  Songs,  etc.,  of  the  U.  S.  Military 
Academy.  By  Lieut.  O.  E.  Wood,  U.  S.  A.  Illus 
trated  by  69  engravings  and  a  copperplate  map. 
Beautifully  printed  on  tinted  paper.  8vo,  cloth 5  oo 

WEST  POINT  LIFE.  A  Poem  read  before  the  Dia 
lectic  Society  of  the  United  States  Military  Academy. 
Illustrated  with  Pen-and-ink  Sketches.  By  a  Cadet 
To  which  is  added  the  song,  "  Benny  Havens,  oh  1" 
oblong  8vo,  21  full  page  illustrations,  cloth 2  50 

GUIDE  TO  WEST  POINT  and  the  U.  S.  Military 
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HENRY.  _  Military  Record  of  Civilian  Appointments  in 
the  United  States  Army,  By  Guy  V.  Henry,  Brevet 
Colonel  and  Captain  First  United  States  Artillery, 
Late  Colonel  and  Brevet  Brigadier  General,  United 
States  Volunteers.  Vol.  i  now  ready.  Vol.  2  in 
press.  8vo,  per  volume,  cloth 5  oo 

HAMERSLY.  Records  of  Living  Officers  of  the  U. 
S.  Navy  and  Marine  Corps.  Compiled  from  official 
sources.  ^  By  Lewis  B.  Hamersly,  late  Lieutenant 
U.  S.  Marine  Corps.  Revised  edition,  8vo,  cloth...  5  oo 

MOORE.  Portrait  Gallery  of  the  War.  Civil,  Military 
and  Naval.  A  Biographical  record,  edited  by  Frank 
Moore.  60  fine  portraits  on  steel.  Royal  8vo, 
cloth 6  oo 

16 


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VAN  NOSTRAND'S  SCIENCE  SERIES, 

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No.  1.— CHIMNEYS  FOR  FURNACES,  FIRE 
PLACES,  AND  STEAM  BOILERS.  By 
R.  ARMSTRONG,  C.  E. 

No.  2.— STEAM  BOILER  EXPLOSIONS.     By  ZE- 

RAIT    COLBURN. 

No.  3.— PRACTICAL  DESIGNING  OF  RETAIN 
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No.  4.- PROPORTIONS  OF  PINS  USED  IN 
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No.  5.— VENTILATION  OF  BUILDINGS.  By  W. 
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